What if you wanted to find the missing sides of a right triangle with angles of
and
and a hypotenuse length of 10 inches? How could you use trigonometry to help you? After completing this Concept, you'll be able to solve problems like this one.

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Guidance

The trigonometric ratios are not dependent on the exact side lengths, but the angles. There is one fixed value for every angle, from
to
. Your scientific (or graphing) calculator knows the values of the sine, cosine and tangent of all of these angles. Depending on your calculator, you should have [SIN], [COS], and [TAN] buttons. Use these to find the sine, cosine, and tangent of any acute angle. One application of the trigonometric ratios is to use them to find the missing sides of a right triangle. All you need is one angle, other than the right angle, and one side.

Concept Problem Revisited

Use trigonometric ratios to find the missing sides. Round to the nearest tenth.

Find the length of
and
using sine or cosine ratios:

Vocabulary

Trigonometry
is the study of the relationships between the sides and angles of right triangles. The legs are called
adjacent
or
opposite
depending on which
acute
angle is being used. The three trigonometric (or trig) ratios are
sine
,
cosine
, and
tangent
.

Guided Practice

1. What is
?

2. Find the length of the missing sides and round your answers to the nearest tenth:
.

3. Find the length of the missing sides and round your answers to the nearest tenth:
.

Answers:

1. Using your calculator, you should find that
?

2. Use tangent for
and cosine for
.

3. Use tangent for
and cosine for
.

Practice

Use your calculator to find the value of each trig function below. Round to four decimal places.

Find the length of the missing sides. Round your answers to the nearest tenth.

Find
and
.

Use your knowledge of where the trigonometric ratios come from to explain your result to the previous question.

Generalize your result to the previous two questions. If
, then
.

How are
and
related? Explain.

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Description

This concept teaches students to find trigonometric ratios with a calculator in order to find missing sides of right triangles.

Learning Objectives

Here you'll learn how to solve for missing sides in right triangles that are not one of the special right triangles.